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Integrable representations of involutive algebras and Ore localization. (English) Zbl 1264.46039

The paper deals with some specific properties of integrable representations of involutive algebras. The analysis is motivated by some well-known problems when dealing with unbounded operators, i.e., the fact that the domains of these operators cannot coincide with the Hilbert space where they are supposed to act.
After a short introductory section on representation theory, meant for non-expert readers, the author deduces some results on the integrability of representations for involutive algebras. Then, for these algebras, the author describes the concept of Ore localization, and shows the existing relation between Ore localization and integrable representations. The paper ends with a concise appendix on basic facts concerning unbounded operators.

MSC:

46K10 Representations of topological algebras with involution
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[1] Antoine, J.P., Inoue, A., Trapani, C.: Partial *-algebras and their Operator Realizations. Kluwer, Boston (2002) · Zbl 1023.46004
[2] Bagarello, F.: Algebras of unbounded operators and physical applications: a survey. Rev. Math. Phys. 19, 231–271 (2007) · Zbl 1144.82301 · doi:10.1142/S0129055X07002961
[3] Bhatt, S.J., Inoue, A., Ogi, H.: Unbounded C*-seminorms and unbounded C*-spectral algebras. J. Oper. Theory 45(1), 53–80 (2001) · Zbl 0994.47074
[4] Borchers, H.J., Yngvason, J.: Partially conmmutative moment problems. Math. Nachr. 145, 111–117 (1990) · Zbl 0733.46044 · doi:10.1002/mana.19901450109
[5] Conway, J.B.: A Course in Functional Analysis. Springer, Berlin Heidelberg New York (1994)
[6] Dubin, D.A., Hennings, M.A.: Quantum Mechanics, Algebras and Distributions. Pitman Research Notes in Mathematics, vol. 238. Longman Scientific and Technical (1990) · Zbl 0705.46039
[7] Fragoulopoulou, M.: Topological Algebras with Involution. North-Holland Mathematical Studies, vol. 200. Elsevier, Amsterdam (2005) · Zbl 1197.46001
[8] Goodearl, K.R., Warfield, R.B.: An Introduction to Noncommutative Noetherian Rings, 2nd edn. Cambridge University Press, Cambridge (2004) · Zbl 1101.16001
[9] Inoue, A.: On a class of unbounded operator algebras. Pac. J. Math. 65(1), 77–95 (1976) · Zbl 0331.46063 · doi:10.2140/pjm.1976.65.77
[10] Inoue, A.: Unbounded representations of symmetric *-algebras. J. Math. Soc. Jpn. 29(2), 219–232 (1977) · Zbl 0356.46054 · doi:10.2969/jmsj/02920219
[11] Inoue, A.: Well-behaved *-representations of *-algebras. Acta Univ. Ouluensis., A Sci. Rerum Nat. 408, 107–117 (2004) · Zbl 1076.46045
[12] Mallios, A., Haralampidou, M.: Topological Algebras and Applications. American Mathematical Society (2005) · Zbl 1109.46003
[13] Powers, R.T.: Self-adjoint algebras of unbounded operators. Commun. Math. Phys. 21, 85–124 (1971) · Zbl 0214.14102 · doi:10.1007/BF01646746
[14] Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser, Cambridge (1990) · Zbl 0697.47048
[15] Schmüdgen, K.: On well-behaved unbounded representations of *-algebras. J. Oper. Theory 48(3), 487–502 (2002) · Zbl 1019.47050
[16] Segal, I.: A non-commutative extension of abstract integration. Ann. Math. 57(3), 401–457 (1953) · Zbl 0051.34201 · doi:10.2307/1969729
[17] Trapani, C.: Unbounded C*-seminorms, biweights, and *-representations of *-algebras: a review. Int. J. Math. Math. Sci. 2006, 1–34 (2006) · Zbl 1138.47054 · doi:10.1155/IJMMS/2006/79268
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